Abstract We demonstrate how quantum walk can simulate exotic cell-like structures for topological phases and boundary states. These cell-like structures contain the three known boundary states of Dirac cone, Fermi arc and flat bands alongside of all trivial and non-trivial phases of BDI family of topological phases. We also characterize the behavior of boundary states through Bloch spheres. In addition, we investigate the topological phase transitions and critical behavior of the system that take place over boundary states through curvature function. We confirm that critical behavior of the simulated topological phenomena can be described by peak-divergence scenario. We extract the critical exponents and length scale, establish a scaling law and show that band crossing is 1. Furthermore, we find the correlation function through Wannier states and show that it decays as a function of length scale.
Abstract We demonstrate how quantum walk can simulate exotic cell-like structures for topological phases and boundary states. These cell-like structures contain the three known boundary states of Dirac cone, Fermi arc and flat bands alongside of all trivial and non-trivial phases of BDI family of topological phases. We also characterize the behavior of boundary states through Bloch spheres. In addition, we investigate the topological phase transitions and critical behavior of the system that take place over boundary states through curvature function. We confirm that critical behavior of the simulated topological phenomena can be described by peak-divergence scenario. We extract the critical exponents and length scale, establish a scaling law and show that band crossing is 1. Furthermore, we find the correlation function through Wannier states and show that it decays as a function of length scale.
Topological gapless phases of matter have been a recent interest among theoretical and experimental condensed matter physicists. Fermionic chains with extended nearest neighbor couplings have been observed to show unique topological transition at the multicritical points between distinct gapless phases. In this work, we show that such topological gapless phases and the transition between them can be simulated in a quantum walk. We consider a three-step discrete-time quantum walk and identify various critical or gapless phases and multicriticalities from the topological phase diagram along with their distinguished energy dispersions. We reconstruct the scaling theory based on the curvature function to study transition between gapless phases in the quantum walk. We show the interesting features observed in fermionic chains, such as diverging, sign flipping and swapping properties of curvature function, can be simulated in the quantum walk. Moreover, the renormalization group flow and Wannier state correlation functions also identify transition at the multicritical points between gapless phases. We observe the scaling law and overlapping of critical and fixed point properties at the multicritical points of the fermionic chains can also be observed in the quantum walk. Furthermore, we categorize the topological transitions at various multicritical points using the group velocity of the energy eigenstates. Finally, the topological characters of various gapless phases are captured using winding number which allows one to distinguish various gapless phases and also show the transitions at the multicritical points.
Topological phases, edge states, and flat bands in synthetic quantum systems are a key resource for topological quantum computing and noise-resilient information processing. We introduce a scheme based on step-dependent quantum walks on cyclic graphs, termed cyclic quantum walks (CQWs), to simulate exotic topological phenomena using discrete Fourier transforms and an effective Hamiltonian. Our approach enables the generation of both gapped and gapless topological phases, including Dirac cone-like energy dispersions, topologically nontrivial flat bands, and protected edge states, all without resorting to resource-intensive split-step or split-coin quantum walk protocols. Odd and even-site cyclic graphs exhibit markedly different spectral characteristics, with rotationally symmetric flat bands emerging exclusively in 4n-site graphs (n∈N). We analytically establish the conditions for the emergence of topological, gapped flat bands and show that gap closings in rotation space imply the formation of Dirac cones in momentum space. Further, we engineer protected edge states at the interface between distinct topological phases in both odd and even cycle graphs. We numerically demonstrate that the edge states are robust against moderate static and dynamic gate disorder as well as remain stable against phase-preserving perturbations and are independent of initial states. This scheme serves as a resource-efficient and versatile platform to engineer topological phases, transitions, edge states, and flat bands in small-scale quantum systems, opening new avenues for robust quantum memory, protected state transfer, and compact implementations of fault-tolerant quantum technologies.
We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We focus mainly on the completion of a table for the protocols of the quantum walk that could simulate different families of the topological phases in one, two, and three dimensions. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences. To further enrich the simulation of the topological phenomena, we include step-dependent coins in the evolution operators of the quantum walks. This leads to step dependence of the simulated topological phenomena and their properties which introduces dynamicity as a feature of simulated topological phases and boundary states. This dynamicity provides the step number of the quantum walk as a means to control and engineer the numbers of topological phases and boundary states, their numbers, types, and even occurrences.
Discrete-time quantum walks are among the branches of quantum information and computation. They are platforms for developing quantum algorithms for quantum computers. In addition, due to their universal primitive nature, discrete-time quantum walks have been used to simulate other quantum systems and phenomena that are observed in physics and chemistry. To fully utilize the potentials that the discrete-time quantum walks hold in their applications, control over the discrete-time quantum walks and their properties becomes essential. In this dissertation, we propose two models for attaining a high level of control over the discrete-time quantum walks. In the first one, we incorporate a dynamical nature for the unitary operator performing the quantum walks. This enables us to readily control the properties of the walker and produce diverse behaviors for it. We show that with our proposal, the important properties of the discrete-time quantum walks such as variance would indeed improve. To explore the potential of this proposal, we apply it in the simulations of topological phases in condensed matter physics. With our proposal, we can control the simulations and determine the type of topological phenomena that should be simulated. In addition, we confirm simulations of topological phases and boundary states that can be observed in one-, two- and three-dimensional systems. Finally, we report the emergence of exotic phase structures in form of cell-like structures that contain all types of topological phases and boundary states of certain classes. In our second proposal, we take advantage of resources available in quantum mechanics, namely quantum entanglement and entangled qubits. In this proposal, we use entangled qubits in the structure of a quantum walk and show that by tuning the initial entanglement between these qubits and how these qubits are modified through the walk, one is able to produce diverse behaviors for the quantum walk and control its behavior.
The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity susceptibility, and the correlation length extracted from the curvature function also gives a momentum scale over which the fidelity susceptibility decays. To map out the profile and criticality of the fidelity susceptibility, we turn to quantum walks that simulate one-dimensional class BDI and two- dimensional class D Dirac models and demonstrate their accuracy in capturing the critical exponents and scaling laws near topological phase transitions.
Topological phase transitions occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and driving, and hence require a variety of techniques and concepts to describe their topological properties. For instance, topology may be accessed from single-particle Bloch wave functions, Green's functions, or many-body wave functions. We demonstrate that despite this diversity, all topological phase transitions display a universal feature: namely, a divergence of the curvature function that composes the topological invariant at the critical point. This feature can be exploited via a renormalization-group–like methodology to describe topological phase transitions. This approach serves to extend notions of correlation function, critical exponents, scaling laws and universality classes used in Landau theory to characterize topological phase transitions in a unified manner.